Example Data Table
| Case |
Sample 1 |
Sample 2 |
Method |
Alpha |
| Class scores |
12, 14, 15, 16, 18, 19 |
10, 11, 13, 14, 15, 17 |
Welch |
0.05 |
| Production batches |
n = 30, mean = 78.4, SD = 8.2 |
n = 28, mean = 73.1, SD = 7.5 |
Pooled |
0.01 |
Formula Used
Mean difference: d = x̄1 - x̄2
Welch standard error: SE = √(s1² / n1 + s2² / n2)
Welch degrees of freedom: df = (a + b)² / [a² / (n1 - 1) + b² / (n2 - 1)], where a = s1² / n1 and b = s2² / n2.
Pooled variance: sp² = [(n1 - 1)s1² + (n2 - 1)s2²] / (n1 + n2 - 2)
Pooled standard error: SE = sp √(1 / n1 + 1 / n2)
Test statistic: t = [(x̄1 - x̄2) - Δ0] / SE
Confidence interval: (x̄1 - x̄2) ± t critical × SE
How to Use This Calculator
Select raw data when you have every value. Select summary statistics when you only have n, mean, and standard deviation.
Choose Welch when variances may differ. Choose pooled only when equal variance is a fair assumption.
Enter alpha, the hypothesized mean difference, and the alternative hypothesis. Then press Calculate.
Read the t statistic, degrees of freedom, p value, confidence interval, and decision. Use CSV or PDF export for records.
Why Two Sample Testing Matters
A two sample t test compares two independent group means. It helps when you want to know whether an observed difference is likely due to chance. This calculator supports raw data and summary statistics. It also supports Welch and pooled variance methods. That makes it useful for lessons, reports, audits, and research notes.
When to Use It
Use this tool when each row belongs to one group only. Common examples include two classes, two machines, two campaigns, or two treatment groups. The measurements should be numeric. The groups should be independent. For paired before and after data, use a paired t test instead.
Choosing the Method
Welch test is the safer default. It does not require equal variances. It adjusts the degrees of freedom using the sample sizes and standard deviations. The pooled test assumes both groups share one population variance. Use it only when that assumption is reasonable. The pooled method can be more powerful when the assumption is true.
Interpreting Results
The t statistic measures the standardized distance between the observed difference and the hypothesized difference. A large absolute t value gives stronger evidence against the null hypothesis. The p value measures how unusual the result would be if the null statement were true. Compare the p value with alpha. A smaller p value leads to rejection.
Confidence and Effects
The confidence interval gives a practical range for the mean difference. If a two sided interval excludes the hypothesized difference, the related two sided test is significant. Effect sizes add practical meaning. Cohen d and Hedges g describe the difference in standard deviation units. Glass delta uses the second group standard deviation.
Good Practice
Always inspect the data before trusting a test. Look for extreme outliers, recording errors, and strong skew. Larger samples make the t method more stable. Small samples need more care. Report the method, t value, degrees of freedom, p value, confidence interval, alpha, and decision. Include the sample means and standard deviations too.
Output Limits
The calculator gives statistical guidance, not final scientific proof. Results depend on valid inputs and assumptions. Use domain knowledge with every decision. Keep original data available for review, replication, and checking by your team.
FAQs
What is a two sample t test?
It is a statistical test that compares the means of two independent groups. It checks whether the observed difference is larger than expected from random sampling variation.
Should I use Welch or pooled?
Use Welch as the usual default. It handles unequal variances better. Use pooled only when equal population variances are reasonable for your data.
What does the p value mean?
The p value shows how unusual the test statistic is under the null hypothesis. A p value below alpha usually leads to rejection of the null statement.
What is alpha?
Alpha is the selected significance level. Common values are 0.05, 0.01, and 0.10. It sets the cutoff for the test decision.
Can I enter summary statistics?
Yes. Select summary statistics. Then enter each sample size, mean, and sample standard deviation. The calculator will use those values directly.
What is the hypothesized difference?
It is the mean difference stated by the null hypothesis. Most tests use zero. A nonzero value tests against a specific expected gap.
What is Cohen d?
Cohen d is an effect size. It expresses the mean difference in pooled standard deviation units. It helps describe practical strength.
Can this test be used for paired data?
No. Paired data needs a paired t test. Use this calculator only when the two groups are independent.